Chow motives associated to certain algebraic Hecke characters

Abstract:

Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character $\lambda _A$ of $FK$ such that $L(A/F,s)=L(\lambda _A,s)$. We consider a certain converse to their result. Namely, let $A$ be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form $y^e=\gamma x^f+\delta$. Fix positive integers $a$ and $n$ such that $n/2 < a \leq n$. Under mild conditions on $e, f, \gamma , \delta$, we construct a Chow motive $M$, defined over $F=\mathbb {Q}(\gamma ,\delta )$, such that $L(M/F,s)$ and $L(\lambda _A^a\overline {\lambda }_A^{n-a},s)$ have the same Euler factors outside finitely many primes.